The Earth-Mover distance is important for clustering problems for the following reason. For any weighted sets \((P,w)\)  and \((B,u)\)  and query \(C\) , \(|\bar{g}(P,w,C) - \bar{g}(B,u,C)| \le d_{EM}(P,w,B,u)\) . This is immediate from a repeated application of the triangle-inequality (proofs are found in Theorem 2.3 of [1]} as well as in [2]}, [3]}, [4]}).
